What is the Todd class *really*? My question is about how to think about the Todd class.
Usually this is presented via Grothendieck Riemann Roch (GRR): if $X$ is a smooth projective scheme over a field $\mathbf{C}$, the chern character does not commute with integration/proper pushforward:
$\require{AMScd}$
\begin{CD}
    K_0(X) @>\int >> K_0(\text{pt})=\mathbf{Z}\\
    @V \text{ch} V V @VV \text{ch} V\\
    \text{H}^*(X) @>\int >> \text{H}^*(\text{pt})=k
\end{CD}
and the Todd class $\text{Td}(\mathcal{T}_X)\in \text{H}^*(X)$ is the correction term to make the above commute. You can then do a computation to explicitly write down what $\text{Td}$ is in terms of chern roots.
I will take this story as the definition of the Todd class. However, on its own the content of the above is just: there exists a correction factor, and here is an explicit formula. It doesn't do a good job of explaining what this correction factor really is, or why it should show up in GRR. Ideally there would be a couple more examples where the Todd class shows up, or ways of understanding the Todd class, so that you could understand the correction factor as "that thing that shows up in $\ldots$".  Notice that the quality of this answer improves the further away these examples are from GRR. For instance, if the folk result about the Todd class being an Euler class had been written down and made precise, it would be a good answer.
My question: in what other contexts (fundamentally different from GRR) does the Todd class appear?
 A: To my knowledge, currently the best "motivation" for the Todd class comes from the so called "orientation theory"  and the formal group laws associated to "oriented" cohomology theories. As far as I know, these ideas go back to Quillen's work on complex cobordism, although current formulation is mainly due to Panin-Smirnov's works (here, here and here). I suggest to read this exposition, which briefly summarizes the main ideas.
Let me summarize the explanation without going into every detail. We say that a cohomology theory $H\colon \mathrm{Var}_k\to \mathrm{Rings}$ is oriented if it has a theory of Chern classes with all usual properties one might expect (invariance under pullback, Withney sum and projective bundle formula, nilpotence...). (I am cheating here, people define orientations differently, but there  is a theorem stating that both things, orientations and reasonable Chern classes, are in fact equivalent under mild conditions). It is worth noticing two facts  about this definition:

*

*the cohomology is not assumed to be graded (so that the $K$-theory is such a cohomology)


*For $L,L'\to X$ two line bundles $c_1(L\otimes L')$ need not to be equal to $c_1(L) + c_1(L')$. Such cohomologies are called additive. For example, $K$-theory satisfies that $c_1(L)=1-[L^*]$ so that $c_1(L\otimes L')=c_1(L) + c_1(L')-c_1(L)\cdot c_1(L')$, which is said to be multiplicative (the name comes from the multiplicative formal group law, where the set of coordinates is taken with respect to 1. That is to say, (1-x)(1-y)=(1-x-y+xy)). The target of the Chern character is always and additive cohomology.
The fun thing is that a given cohomology theory $H$ may have different orientations, in other words, different theories of Chern classes. However two orientations can be related concretely: if $H$ is oriented, with some first Chern classes $c_1$, and we have a new orientation, with a new first Chern classes $c_1^{\mathrm{new}}$, then it is easy to check that
$$
c_1^{\mathrm{new}}(L)=F(c_1(L))\cdot c_1(L)
$$
where $F(t)\in H(pt)[[t]]$ is an invertible series and with coefficients in the cohomology ring of a point. This $F$ is the (inverse) Todd series associated to the new orientation and can be applied, through the usual multiplicative extension, to any virtual vector bundle.
Let $\varphi \colon H\to H'$  be a morphism of cohomology theories (i.e., natural transformation between contravariant functor of rings) it is also easy to check that the orientation on $H$ provides an orientation on $H'$ or, in other words, new Chern classes on $H'$ and therefore, an associated (inverse) Todd series.
Main example: Consider the Chern character $\mathrm{ch}\colon K\to H_{\mathbb{Q}}$, which is a morphism of cohomologies where $H_{\mathbb{Q}}$ is additive. Let $c_1^K$  denote the Chern classes with values in $K$-theory, to avoid confussion. Let us check the associated (inverse) Todd series to $\mathrm{ch}$:
$$
\mathrm{ch}(c^K_1(L))=\mathrm{ch}(1-[L^*]))=1-e^{c_1(L^*)}=1-e^{-c_1(L)}=\frac{1-e^{-c_1(L)}}{c_1(L)}\cdot c_1(L)
$$
Hence, its associated series is $\frac{1-e^{-t}}{t}$, which is precisely the inverse of the classical Todd series.
Now, to concretely answer your question, here you have:
Other examples: The universal map from cobordsim to any oriented cohomology theory $\Omega\to H$ also has its associated (inverse) Todd series, but it is trivial (the series is $F(t)=1$). However, the associated Riemann-Roch type theorem is not at all trivial, is the so-called Conner-Floyd theorem.
On any oriented cohomology theory $H$ you can consider  a new theory of Chern classes: $c_1^\mathrm{new}(L):=-c_1(L^*)$,  its associated (inverse) Todd series is $F(t)=1$  if $H$ is additive, but it is the formal inverse otherwise.
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In the aformentioned exposition you have these kind of arguments with details and "without cheating" (orientations are defined properly, and not just as a "theory of Chern classes"). The Riemann-Roch theorem in this context is as  corollary of the universal property of $K$-theory (which is the universal multiplicative oriented cohomology theory).
